Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

Abstract

We prove an $$\varepsilon $$ -regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $$L^2$$ -norm. Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $${\mathbb {S}}^{4,1}$$ .